Advertisement

Russian Journal of Mathematical Physics

, Volume 17, Issue 4, pp 495–508 | Cite as

Special functions related to Dedekind-type DC-sums and their applications

  • Y. SimsekEmail author
Article

Abstract

In this paper, we construct trigonometric functions in the form of a sum T p (h, k) which is referred to as a Dedekind-type DC-(Dahee and Changhee) sum. We establish analytic properties of this sum, find its trigonometric representations, and prove a reciprocity theorem for these sums. Furthermore, we obtain relationships between the Clausen functions, polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), Hardy-Berndt sums, and the sum T p (h, k). We also give some applications related to these sums and functions.

Keywords

Zeta Function Euler Polynomial Euler Function Hurwitz Zeta Function Bernoulli Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Applied Mathematics Series-55, 1965).Google Scholar
  2. 2.
    T. M. Apostol, “Generalized Dedekind Sums and Transformation Formulae of Certain Lambert Series,” Duke Math. J. 17, 147–157 (1950).CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    T. M. Apostol, “On the Lerch Zeta Function,” Pacific J. Math. 1, 161–167 (1951).MathSciNetzbMATHGoogle Scholar
  4. 4.
    T. M. Apostol, “Theorems on Generalized Dedekind Sums,” Pacific J. Math. 2, 1–9 (1952).MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Bayad, “Sommes elliptiques multiples d’Apostol-Dedekind-Zagier,” Comptes Rendus Math. 339(7), 457–462 (2004).CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Beck, “Dedekind Cotangent Sums,” Acta Arith. 109(2), 109–130 (2003).CrossRefMathSciNetADSzbMATHGoogle Scholar
  7. 7.
    B. C. Berndt, “On the Hurwitz Zeta-Function,” Rocky Mountain J. Math. 2(1), 151–157 (1972).CrossRefMathSciNetADSzbMATHGoogle Scholar
  8. 8.
    B. C. Berndt, “Dedekind Sums and a Paper of G. H. Hardy,” J. London Math. Soc. (2) 13(1), 129–137 (1976).CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    B. C. Berndt, “Reciprocity Theorems for Dedekind Sums and Generalizations,” Advances in Math. 23(3), 285–316 (1977).CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    B. C. Berndt, “Analytic Eisenstein Series, Theta Functions and Series Relations in the Spirit of Ramanujan,” J. Reine Angew. Math. 303/304, 332–365 (1978).MathSciNetGoogle Scholar
  11. 11.
    B. C. Berndt and L. A. Goldberg, “Analytic Properties of Arithmetic Sums Arising in the Theory of the Classical Theta Functions,” SIAM J. Math. Anal. 15(1), 143–150 (1984).CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    B. C. Berndt and B. P. Yeap, “Explicit Evaluations and Reciprocity Theorems for Finite Trigonometric Sums,” Adv. in Appl. Math. 29(3), 358–385 (2002).CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    M. Can, M. Cenkci, and V. Kurt, “Generalized Hardy-Berndt Sums,” Proc. Jangjeon Math. Soc. 9(1) 19–38 (2006).MathSciNetzbMATHGoogle Scholar
  14. 14.
    E. Carneiro, “Sharp Approximations to the Bernoulli Periodic Functions by Trigonometric Polynomials,” J. Approx. Theory 154(2), 90–104 (2008).CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    M. Cenkci, Y. Simsek, M. Can, and V. Kurt, “Twisted Dedekind Type Sums Associated with Barnes’ Type Multiple Frobenius-Euler l-Functions,” Advanc. Stud. Contemp. Math. 18(2), 135–160 (2009), arXiv:0711.0579v1 [math.NT].MathSciNetzbMATHGoogle Scholar
  16. 16.
    D. Cvijovic, “Integral Representations of the Legendre Chi Function,” J. Math. Anal. Appl. 332, 1056–1062 (2007).CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    D. Cvijovic and J. Klinowski, “Values of the Legendre Chi and Hurwitz Zeta Functions at Rational Arguments,” Math. Comp. 68, 1623–1630 (1999).CrossRefMathSciNetADSzbMATHGoogle Scholar
  18. 18.
    J. Choi, “Some Identities Involving the Legendre’s Chi-Function,” Commun. Korean Math. Soc. 22(2), 219–225 (2007).CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    J. Choi, D. S. Jang, and H. M. Srivastava, “A Generalization of the Hurwitz-Lerch Zeta Function,” Integral Transforms Spec. Funct. 19(1–2), 65–79 (2008).MathSciNetzbMATHGoogle Scholar
  20. 20.
    J. Choi, H. M. Srivastava, and V. S. Adamchik, “Multiple Gamma and Related Functions,” Appl. Math. Comput. 134, 515–533 (2003).CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    R. Dedekind, “Erlauterungen zu zwei Fragmenten von Riemann,” Bernhard Riemann’s Gesammelte Mathematische Werke, 2nd ed. (B. G. Teubner, Leipzig, 1892), pp. 466–472.Google Scholar
  22. 22.
    U. Dieter, “Cotangent Sums, a Further Generalization of Dedekind Sums,” J. Number Theory 18(3), 289–305 (1984).CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    O. Espinosa and V. H. Moll, “On Some Integral Involving the Hurwitz Zeta Function: Part 2,” Ramanujan J. 6, 449–468 (2002).CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    L. A. Goldberg, Transformations of Theta Functions and Analogues of Dedekind Sums, PhD Thesis (Vassar College, Urbana, Illinois, 1975).Google Scholar
  25. 25.
    E. Grosswald, “Dedekind-Rademacher Sums,” Amer. Math. Monthly 78, 639–644 (1971).CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    E. Grosswald and H. Rademacher, Dedekind Sums, Carus Monogr., no. 16 (Math. Assoc. Amer., Washington, D. C., 1972).Google Scholar
  27. 27.
    J. Guillera, J. Sondow, “Double Integrals and Infinite Products for Some Classical Constants via Analytic Continuations of Lerch’s Transcendent,” Ramanujan J. 16, 247–270 (2008), arXiv:math/0506319v3 [math.NT].CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    M. E. Hoffman, “Derivative Polynomials and Associated Integer Sequences,” Electronic J. Combinatorics 6, #R21 (1999).Google Scholar
  29. 29.
    S. Iseki, “The Transformation Formula for the Dedekind Modular Function and Related Functional Equation,” Duke Math. J. 24, 653–662 (1957).CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    T. Kim, “A Note on p-Adic q-Dedekind Sums,” C. R. Acad. Bulgare Sci. 54(10), 37–42 (2001).MathSciNetzbMATHGoogle Scholar
  31. 31.
    T. Kim, “q-Volkenborn Integration,” Russ. J. Math. Phys. 9(3), 288–299 (2002).MathSciNetzbMATHGoogle Scholar
  32. 32.
    T. Kim, “q-Extension of the Euler Formula and Trigonometric Functions,” Russ. J. Math. Phys. 14(3), 275–278 (2007).CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    T. Kim, “q-Euler Numbers and Polynomials Associated with p-Adic q-Integrals,” J. Nonlinear Math. Phys. 14(1), 15–27 (2007).CrossRefMathSciNetADSzbMATHGoogle Scholar
  34. 34.
    T. Kim, “Euler Numbers and Polynomials Associated with Zeta Functions,” Abstract and Applied Analysis, vol. 2008, Article ID 581582, 11 pages, 2008. doi:10.1155/2008/581582, arXiv:0801.0329v1 [math.NT].Google Scholar
  35. 35.
    T. Kim, “Note on the Euler Numbers and Polynomials,” Adv. Stud. Contemp. Math. 17(2), 109–116 (2008).MathSciNetGoogle Scholar
  36. 36.
    T. Kim, “On p-Adic Interpolating Function for q-Euler Numbers and Its Derivatives,” J. Math. Anal. Appl. 339(1), 598–608 (2008).CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    T. Kim, “The Modified q-Euler Numbers and Polynomials,” Adv. Stud. Contemp. Math. (Kyungshang) 16(2), 161–170 (2008).MathSciNetzbMATHGoogle Scholar
  38. 38.
    T. Kim, “Note on the Euler Numbers and Polynomials,” Adv. Stud. Contemp. Math. (Kyungshang) 17(2), 31–136 (2008).Google Scholar
  39. 39.
    T. Kim, “Note on Dedekind type DC sums,” Adv. Stud. Contemp. Math. 18(2), 249–260 (2009), arXiv:0812.2553v1[math.NT].MathSciNetzbMATHGoogle Scholar
  40. 40.
    T. Kim, Y. Kim, and K. Hwang, “On the q-Extensions of the Bernoulli and Euler Numbers,” Related Identities and Lerch Zeta Function, arXiv:0901.0249v1 [math.NT].Google Scholar
  41. 41.
    V. Kurt, “On Dedekind Sums,” Indian J. Pure Appl. Math. 21(10), 893–896 (1990).MathSciNetzbMATHGoogle Scholar
  42. 42.
    M. A. Lerma, “The Bernoulli Periodic Functions,” www.math.northwestern.edu/~mlerma/papers.
  43. 43.
    H. Ozden and Y. Simsek, “A New Extension of q-Euler Numbers and Polynomials Related to Their Interpolation Functions,” Appl. Math. Letters, 21(9), 934–939 (2008).CrossRefMathSciNetGoogle Scholar
  44. 44.
    H. Rademacher, “Über eine Reziprozitätsformel aus der Theorie der Modulfunktionen,” Mat. Fiz. Lapok 40, 24–34 (1933) [in Hungarian].zbMATHGoogle Scholar
  45. 45.
    H. Rademacher, “Die Reziprozitatsformel für Dedekindsche Summen,” Acta Sci. Math. (Szeged) 12(B), 57–60 (1950).MathSciNetGoogle Scholar
  46. 46.
    H. Rademacher, Topics in Analytic Number Theory, Grundlehren Math. Wiss. 169 (Springer-Verlag, Berlin, 1973).zbMATHGoogle Scholar
  47. 47.
    G. Rządkowski and S. Łepkowski, “A Generalization of the Euler-Maclaurin Summation Formula: An application to numerical computation of the Fermi-Dirac integrals,” J. Sci. Comput. 35(1), 63–74 (2008).CrossRefMathSciNetzbMATHGoogle Scholar
  48. 48.
    G. H. Hardy, “On Certain Series of Discontinuous Functions Connected with the Modular Functions,” Quart. J. Math. 36, 93–123 (1905) = Collected Papers, vol. IV, pp. 362–392 (Clarendon Press Oxford, 1969).Google Scholar
  49. 49.
    Y. Simsek, “Relation between Theta Function Hardy Sums Eisenstein and Lambert Series in the Transformation Formula of log η g,h(z),” J. Number Theory 99(2), 338–360 (2003).CrossRefMathSciNetzbMATHGoogle Scholar
  50. 50.
    Y. Simsek, “Generalized Dedekind Sums Associated with the Abel Sum and the Eisenstein and Lambert Series,” Adv. Stud. Contemp. Math. 9(2), 125–137 (2004).MathSciNetzbMATHGoogle Scholar
  51. 51.
    Y. Simsek, “On Generalized Hardy’s Sums s 5(h, k),” Ukrain. Mat. Zh. 56(10), 1434–1440 (2004) [Ukrainian Math. J. 56 (10), 1712–1719 (2004) (2005)].MathSciNetzbMATHGoogle Scholar
  52. 52.
    Y. Simsek, “q-Analogue of the Twisted l-Series and q-Twisted Euler Numbers,” J. Number Theory 110(2), 267–278 (2005).CrossRefMathSciNetzbMATHGoogle Scholar
  53. 53.
    Y. Simsek, “q-Dedekind Type Sums Related to q-Zeta Function and Basic L-Series,” J. Math. Anal. Appl. 318(1), 333–351 (2006).CrossRefMathSciNetzbMATHGoogle Scholar
  54. 54.
    Y. Simsek, “p-Adic q-Higher-Order Hardy-Type Sums,” J. Korean Math. Soc. 43(1), 111–131 (2006).CrossRefMathSciNetzbMATHGoogle Scholar
  55. 55.
    Y. Simsek, “Twisted (h, q)-Bernoulli Numbers and Polynomials Related to Twisted (h, q)-Zeta Function and L-Function,” J. Math. Anal. Appl. 324(2), 790–804 (2006).CrossRefMathSciNetzbMATHGoogle Scholar
  56. 56.
    Y. Simsek, “Generating Functions of the Twisted Bernoulli Numbers and Polynomials Associated with Their Interpolation Functions,” Adv. Stud. Contemp. Math 16(2), 251–278 (2008).MathSciNetzbMATHGoogle Scholar
  57. 57.
    Y. Simsek, “On Analytic Properties and Character Analogs of Hardy Sums,” Taiwanese J. Math. 13(1), 253–268 (2009).MathSciNetzbMATHGoogle Scholar
  58. 58.
    Y. Simsek, “q-Hardy-Berndt Type Sums Associated with q-Genocchi Type Zeta and q-l-Functions,” Nonlinear Anal. 71(12), e377–e395 (2008).MathSciNetGoogle Scholar
  59. 59.
    Y. Simsek, D. Kim, and J. K. Koo, “On Elliptic Analogue of the Hardy Sums,” Bull. Korean Math. Soc. 46(1), 1–10 (2009).CrossRefMathSciNetzbMATHGoogle Scholar
  60. 60.
    H. M. Srivastava, “A Note on the Closed-Form Summation of Some Trigonometric Series,” Kobe J. Math. 16(2), 177–182 (1999).MathSciNetzbMATHGoogle Scholar
  61. 61.
    H. M. Srivastava, “Some Formulas for the Bernoulli and Euler Polynomials at Rational Arguments,” Math. Proc. Cambridge Philos. Soc. 129(1), 77–84 (2000).CrossRefMathSciNetADSzbMATHGoogle Scholar
  62. 62.
    H. M. Srivastava and A. Pinter, “Remarks on Some Relationships between the Bernoulli and Euler Polynomials,” Appl. Math. Lett. 17(4), 375–380 (2004).CrossRefMathSciNetzbMATHGoogle Scholar
  63. 63.
    H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, (Kluwer Academic Publishers, Dordrecht, Boston and London, 2001).zbMATHGoogle Scholar
  64. 64.
    R. Sitaramachandrarao, “Dedekind and Hardy Sums,” Acta Arith. 48, 325–340 (1987).MathSciNetzbMATHGoogle Scholar
  65. 65.
    S. K. Suslov, “Some Expansions in Basic Fourier Series and Related Topics,” J. Approx. Theory 115(2), 289–353 (2002).CrossRefMathSciNetzbMATHGoogle Scholar
  66. 66.
    S. B. Tričković, M. V. Vidanović, and M. S. Stanković, “On the Summation of Trigonometric Series,” Integral Transform Spec. Funct. 19(6), 441–452 (2008).CrossRefzbMATHGoogle Scholar
  67. 67.
    D. H. Trahan, “Regions of Convergence for a Generalized Lambert Series,” Math. Mag. 54(1), 28–32 (1981).CrossRefMathSciNetzbMATHGoogle Scholar
  68. 68.
    R. Dedekind, Erlauterungen zu den Fragmenten XXVIII, in: B. Riemann, Gesammelte mathematische Werke und wissenschaftlicher Nachlass (Dover, New York, N.Y., 1953), pp. 466–478.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Faculty of Arts and Science, Department of MathematicsAkdeniz UniversityAntalyaTurkey

Personalised recommendations